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1.
Diego Maldonado 《Potential Analysis》2016,44(1):169-188
It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampère quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampère equation under minimal geometric assumptions. 相似文献
2.
Luis A. Caffarelli & Yu Yuan 《分析论及其应用》2022,38(2):121-127
We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side. 相似文献
3.
This paper deals with some parabolic Monge-Ampère equation raised from mathematical finance: V_sV_(yy)+ryV_yV_(yy)-θV_y~2= 0(V_(yy) 0). The existence and uniqueness of smooth solution to its initial-boundary value problem with some requirement is obtained. 相似文献
4.
We consider the following initial-boundary-value problem of nonlinear parabolic equations Let h denote step size of space and τ denote step size of time. We use the following notation We give the difference scheme for the problem (1)-(2) 相似文献
5.
Let (W,,H) be an abstract Wiener space assume two
i
,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between 1 and 2 with respect to the Cameron-Martin space
to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are 1 and 2. In this case we prove the existence of a unique (cyclically monotone) map T=I
W
+, with :WH, such that T maps 1 to 2. Moreover, if 2, then T is stochastically invertible, i.e., there exists S:WW such that ST=I
W
1 a.s. and TS=I
W
2 a.s. If, in addition, 1=, then there exists a 1-convex function in the Gaussian Sobolev space
such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W.
Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about 1 and 2.In fact this hypothesis is too strong, cf. Theorem 6.1.
AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks. 相似文献
6.
Yanyan Li & Siyuan Lu 《分析论及其应用》2022,38(2):128-147
We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a
positive bounded periodic function. We prove that $u$ must be the sum of a quadratic
polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named
author. 相似文献
7.
By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the a... 相似文献
8.
For the more general parabolic Monge-Ampère equations defined by the
operator $F(D^2u + σ(x))$, the existence and uniqueness of the admissible solution to
the third initial-boundary value problem for the equation are established. A new
structure condition which is used to get a priori estimate is established. 相似文献
9.
Erhard Heinz Ralf Beyerstedt 《Calculus of Variations and Partial Differential Equations》1994,2(2):241-247
Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s
2=E in the punctured disk 0<(x–x
0)2+(y–y
0)2<2. Assume thatq is continuous at (x0, y0). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x
0,y
0) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344). 相似文献
10.
For the more general parabolic Monge-Ampère equations defined by the operator F(D2u + σ(x)),the existence and uniqueness of the admissible solution to the third initial-boundary value problem for the e... 相似文献
11.
In this paper, we consider the complex Monge-Ampère equation posed on a compact K?hler manifold. We show how to get L~p(p ∞) and L∞estimates for the gradient of the solution in terms of the continuity of the right-hand side. 相似文献
12.
John Urbas 《Mathematische Zeitschrift》2001,236(3):625-641
We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending
on for some .
Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000 相似文献
13.
The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic
Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ
n
. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2]. 相似文献
14.
15.
In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions. 相似文献
16.
Ph. Delanoë 《manuscripta mathematica》1983,45(1):29-45
In [4] (Theorem 2), we solved a Monge-Ampère equation, on a compact Riemannian manifold, which isnot a priori locally invertible. In particular, one cannot expect the solution to be unique. We investigate here, the way bifurcation phenomena may occur in such a case, with some explicit computations on flat tori. 相似文献
17.
This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD2u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in RN with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C∞(Ω) is positive in Ω, but may be appropriate singular on the boundary. 相似文献
18.
Let (X, ω) be a compact Kähler manifold. We introduce and study the largest set DMA(X, ω) of ω-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(X, ω) of all ω-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all ω-psh functions. As a consequence, any ω-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy. 相似文献
19.
Alessandro De Paris Alexandre M. Vinogradov 《Central European Journal of Mathematics》2011,9(4):731-751
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect
to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants
is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce
a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants
of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an
example we study equations of the form u
xy
+ f(x, y, u
x
, u
y
) = 0 and in particular find a simple linearization criterion. 相似文献
20.
We first define molecules for Hardy spaces
H1F(\mathbbRn)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}) associated with a family F\mathcal{F} of sections which is closely related to the Monge-Ampère equation and prove their molecular characters. As an application,
we show that Monge-Ampère singular operators are bounded on
H1F(\mathbbRn)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}). 相似文献